Let $\tau_n$ localize the nonnegative local martingale $M$. For $s\leq t$,
$$
\mathbb E(M_{t\wedge\tau_n}\mid\mathcal F_s)=M_{s\wedge\tau_n}.
$$
<Conditional Fatou lemma> and nonnegativity give
$$
\mathbb E(M_t\mid\mathcal F_s)
\leq\liminf_nM_{s\wedge\tau_n}=M_s.
$$
Thus $M$ is a <supermartingale>, recovering the general fact about a <nonnegative local martingale>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / b / Solution

For a continuous local martingale $M$ with $M_0=0$, its <stochastic exponential> is
$$
\mathcal E(M)_t=\exp\!\left(M_t-\frac12[M]_t\right).
$$
The <Itô formula> gives $d\mathcal E(M)_t=\mathcal E(M)_t\,dM_t$, so $\mathcal E(M)$ is a positive <continuous local martingale>. Every <nonnegative local martingale> is a <supermartingale>, because localization and the <Conditional Fatou lemma> turn the localized martingale equality into the supermartingale inequality.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 3 / a / Solution

{c}

For nonnegative random variables $X_n$ and a sigma-algebra $\mathcal G$,
$$
\mathbb E[\liminf_nX_n\mid\mathcal G]
\leq\liminf_n\mathbb E[X_n\mid\mathcal G]
$$
almost surely. It follows from the ordinary <Fatou lemma> after testing both sides on every event in $\mathcal G$.

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 Incoming links: Conditional Fatou lemma